WASHINGTON — For weeks in late 2020, Senator Mike Lee, Republican of Utah, cheered on President Donald J. Trump’s effort to fight his election defeat, privately offering up “a group of ready and loyal advocates who will go to bat for him.”

In texts to Mark Meadows, then the White House chief of staff, Mr. Lee encouraged the Trump campaign to embrace Sidney Powell, a pro-Trump lawyer whom the senator described as a “straight shooter,” and said the president should “hire the right legal team and set them loose immediately.”

But when Ms. Powell put forth wild claims of foreign rigging of election machines at a widely derided news conference in November, Mr. Lee was chagrined and quietly began to question what Mr. Trump was up to.

“I’m worried about the Powell press conference,” Mr. Lee wrote in another text message to Mr. Meadows. “The potential defamation liability for the president is significant here.”

That message and several others from Mr. Lee, as well as a separate set of exchanges between Representative Chip Roy, Republican of Texas, and Mr. Meadows, trace an about-face by the two Republican lawmakers. The pair started out as enthusiastic supporters of Mr. Trump’s claims of a stolen election but gradually grew alarmed about his push to invalidate the results and ultimately opposed his bid to get Congress to overturn them on Jan. 6, 2021.

The text messages, which are in the possession of the House committee investigating the Capitol riot, were obtained by CNN and authenticated by The New York Times.

They provide a window into the eagerness of Republicans — even some who ended up voting on Jan. 6 to confirm Joseph R. Biden Jr.’s victory — to believe Mr. Trump’s false claims of widespread fraud and their willingness to go to great lengths, including attempts at exploiting the nation’s election laws, to keep him in power. They also illustrate how rapidly those efforts spiraled out of control, and they show a keen awareness on the part of at least some Republicans involved that the endeavor had become untenable to the point of being dangerous.

The text messages were sent to and from Mr. Meadows, who turned them over to the House committee while he was cooperating with the panel. Mr. Meadows later refused to sit for an interview with the committee, and the House voted to recommend that the Justice Department prosecute him for criminal contempt of Congress.

A lawyer for Mr. Meadows did not respond to a request for comment. A spokesman for the committee declined to comment.

The text messages with Mr. Meadows show that Mr. Lee tried several times to offer advice and support for the effort to overturn the election, using multiple strategies.

Mr. Lee suggested that Mr. Trump should “dissociate himself” from Ms. Powell’s false claims after her performance at the November news conference, but even after that, the senator vouched for the conservative lawyer John Eastman, who wrote a memo outlining plans for overturning the election that members of both parties have likened to a blueprint for a coup.

Mr. Lee then endorsed a plan to have legislatures in “a very small handful of states” that Mr. Biden had won put forth pro-Trump electors, as part of a scheme proposed by Mr. Eastman to allow Vice President Mike Pence to reject Mr. Biden’s victory.

But Mr. Lee backed off the effort after no state legislature convened to certify so-called alternate electors, and he began criticizing plans by Senators Ted Cruz of Texas and Josh Hawley of Missouri, both Republicans, to use Congress’s official count of electoral votes on Jan. 6 to challenge the election outcome.

“I have grave concerns with the way my friend Ted is going about this effort,” Mr. Lee wrote to Mr. Meadows.

Mr. Lee ultimately voted to confirm Mr. Biden’s victory. More than half of Republicans in Congress — eight senators and 139 House members — voted to invalidate it, after a mob of Mr. Trump’s supporters, enraged by the lie of a stolen election, stormed the Capitol demanding that it be overturned.

A spokesman for Mr. Lee confirmed the authenticity of the text messages and said they told “the same story Senator Lee told from the floor of the Senate the day he voted to certify the election results of each and every state in the nation.”

“They tell the story of a U.S. senator fulfilling his duty to Utah and the American people by following the Constitution,” the spokesman, Lee Lonsberry, said, citing the senator’s remarks after the deadly riot, which injured more than 150 police officers.

Mr. Roy’s text messages with Mr. Meadows tell a similar tale of a lawmaker who appeared eager to fight alongside Mr. Trump but ultimately backed off when evidence of a stolen election did not appear.

“Dude, we need ammo,” Mr. Roy wrote to Mr. Meadows on Nov. 7, before the Texas lawmaker traveled to Georgia to try to assist in the effort to fight that state’s election results. “We need fraud examples. We need it this weekend.”

But Mr. Roy also warned against making inflammatory claims without evidence.

“We must urge the President to tone down the rhetoric, and approach the legal challenge firmly, intelligently and effectively without resorting to throwing wild desperate haymakers, or whipping his base into a conspiracy frenzy,” Mr. Roy wrote on Nov. 9.

The text messages show that Mr. Roy was also initially supportive of Mr. Eastman’s efforts but grew more skeptical as weeks went by and evidence of widespread fraud failed to materialize.

“The President should call everyone off,” Mr. Roy wrote to Mr. Meadows on Dec. 31. “It’s the only path.”

“If we substitute the will of states through electors with a vote by Congress every 4 years,” he added, “we have destroyed the electoral college.”

The next day, Mr. Roy followed up. If Mr. Trump “allows this to occur,” he wrote to Mr. Meadows, “we’re driving a stake in the heart of the federal republic.”

On Jan. 6, the day that rioters stormed the Capitol, Mr. Roy again reached out to Mr. Meadows.

“This is a sh*tshow,” he wrote. “Fix this now.”

Mr. Meadows replied: “We are.”

A spokesman for Mr. Roy said the congressman’s texts spoke for themselves.